Logic of approximate entailment in quasimetric and in metric spaces

Abstract

It is known that a quasimetric space can be represented by means of a metric space; the points of the former space become closed subsets of the latter one, and the role of the quasimetric is assumed by the Hausdorff quasidistance. In this paper, we show that, in a slightly more special context, a sharpened version of this representation theorem holds. Namely, we assume a quasimetric to fulfil separability in the original sense due to Wilson. Then any quasimetric space can be represented by means of a metric space such that distinct points are assigned disjoint closed subsets.This result is tailored to the solution of an open problem from the area of approximate reasoning. Following the lines of E. Ruspini’s work, the Logic of Approximate Entailment () is based on a graded version of the classical entailment relation. We present a proof calculus for and show its completeness with regard to finite theories.

On Ruspini’s Models of Similarity-Based Approximate Reasoning

“…In a series of papers [7,[16][17][18], the authors have been concerned with logics to reason about graded entailments. Graded approximate and strong entailments are taken as primitive objects of a propositional language.…”

“…In [16,17], we have proposed a simplified proof system for a variant of LAE. Namely, we have focused on the case of ⊗-similarity relations, where ⊗ is the product t-norm.…”

“…In [16], we have shown a similar statement but without the assumption that the similarity relation is symmetric, and we have represented proofs by weighted directed forests. In [17], we have established that spaces based on a possibly non-symmetric similarity relation can, in a certain sense, be embedded into a space based on a similarity relation in the usual sense. Both results combined lead to the completeness theorem mentioned.…”